$A_{\text{rechtw. Dreieck}}={\displaystyle \frac{\left({\text{Kathete}}_{kl}\right)\cdot \left({\text{Kathete}}_{gr}\right)}{2}}\backslash \; A\_\{\backslash text\{rechtw.\; Dreieck\}\}\backslash \; =\backslash \; \backslash dfrac\{\backslash left(\backslash text\{Kathete\}\_\{kl\}\backslash right)\backslash cdot\backslash left(\backslash text\{Kathete\}\_\{gr\}\backslash right)\}\{2\}\backslash$

${\text{Kathete}}_{kl}=\sqrt{17^2-15^2}=\sqrt{289-225}=8\backslash \; \backslash text\{Kathete\}\_\{kl\}\backslash \; =\backslash \; \backslash sqrt\{17^2-15^2\}=\backslash sqrt\{289-225\}\backslash \; =\backslash \; 8\backslash$

${\text{Kathete}}_{kl}=8\text{ cm}\backslash \; \backslash text\{Kathete\}\_\{kl\}\backslash \; =\backslash \; 8\backslash \; \backslash text\{cm\}$

$A_{\text{rechtw. Dreieck}}={\displaystyle \frac{8\cdot 15}{2}}=60{\text{cm}}^2\backslash \; A\_\{\backslash text\{rechtw.\; Dreieck\}\}=\backslash dfrac\{8\backslash cdot15\}\{2\}\backslash \; =\backslash \; 60\backslash \; \backslash text\{cm\}^2$

Berechne das Volumen des Dreiecksprismas:

$V_{\text{Dreieckspr.}}=A_{\text{rechtw. Dreieck}}\cdot h\Rightarrow V_{\text{Dreieckspr.}}=60\cdot 13=780{\text{cm}}^3\backslash \; V\_\{\backslash text\{Dreieckspr.\}\}\backslash \; =\backslash \; A\_\{\backslash text\{rechtw.\; Dreieck\}\}\backslash cdot\; h\backslash Rightarrow\; V\_\{\backslash text\{Dreieckspr.\}\}\backslash \; =\backslash \; 60\backslash cdot\backslash \; 13\backslash \; =\backslash \; 780\backslash \; \backslash text\{cm\}^3$

Berechne das Volumen des Halbzylinders:

$V_{\text{Halbzyl.}}={\displaystyle \frac{{\text{r}}^2\cdot \pi \cdot h}{2}}\Rightarrow V_{\text{Halbzyl.}}={\displaystyle \frac{{\text{3}}^2\cdot 3,14\cdot 13}{2}}\backslash \; V\_\{\backslash text\{Halbzyl.\}\}\backslash \; =\backslash \; \backslash dfrac\{\backslash text\{r\}^2\backslash cdot\; \backslash pi\backslash cdot\; h\}\{2\}\backslash \; \backslash Rightarrow\backslash \; \backslash \; V\_\{\backslash text\{Halbzyl.\}\}\backslash \; =\backslash \; \backslash dfrac\{\backslash text\{3\}^2\backslash cdot\; 3\{,\}14\backslash cdot\; 13\}\{2\}$

$V_{\text{Halbz.}}=183,69{\text{cm}}^3\backslash \; V\_\{\backslash text\{Halbz.\}\}\backslash \; =\backslash \; 183\{,\}69\backslash \; \backslash text\{cm\}^3$

Hinweis: in dieser Rechnung wurde mit dem Näherungswert $3,143\{,\}14$ für $\pi \backslash pi$ gerechnet.

Wenn du am Taschenrechner die $\pi \backslash pi$-Taste benutzt, erhältst du einen etwas abweichenden Wert fur$V_{\text{Halbzyl.}}\backslash \; V\_\{\backslash text\{Halbzyl.\}\}$, nämlich: $183,78{\text{cm}}^3\backslash \; 183\{,\}78\backslash text\{cm\}^3$.

$V_{\text{Ges.}}=V_{\text{Dreieckspr.}}-V_{\text{Halbzyl.}}\Rightarrow V_{\text{Ges.}}=780-183,69\backslash \; V\_\{\backslash text\{Ges.\}\}\backslash \; =\backslash \; V\_\{\backslash text\{Dreieckspr.\}\}-V\_\{\backslash text\{Halbzyl.\}\}\backslash \; \backslash Rightarrow\backslash \; V\_\{\backslash text\{Ges.\}\}=780-183\{,\}69$

$\underset{‾}{\underset{‾}{V_{\text{Ges.}}=596,31{\text{cm}}^3}}\backslash \; \backslash underline\{\backslash underline\{V\_\{\backslash text\{Ges.\}\}\backslash \; =\backslash \; 596\{,\}31\backslash \; \backslash text\{cm\}^3\}\}$

Das Volumen des abgebildeten Körpers beträgt:$596,31{\text{cm}}^3\backslash \; 596\{,\}31\backslash \; \backslash text\{cm\}^3$

Das Volumen des Dreiecksprismas berechnet man mit:

$V_{\text{Dreieckspr.}}=A_{\text{rechtw. Dreieck}}\cdot h\backslash \; V\_\{\backslash text\{Dreieckspr.\}\}\backslash \; =\backslash \; A\_\{\backslash text\{rechtw.\; Dreieck\}\}\backslash cdot\; h$

Wird nun die Kantenlänge $hh$ verdoppelt, wird auch das Volumen des Dreiecksprismas verdoppelt.